Integrand size = 19, antiderivative size = 476 \[ \int \frac {x^3 \sin (c+d x)}{\left (a+b x^2\right )^3} \, dx=-\frac {d x \cos (c+d x)}{8 b^2 \left (a+b x^2\right )}+\frac {3 d \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 \sqrt {-a} b^{5/2}}-\frac {3 d \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 \sqrt {-a} b^{5/2}}-\frac {d^2 \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right ) \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 b^3}-\frac {d^2 \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right ) \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{16 b^3}-\frac {x^2 \sin (c+d x)}{4 b \left (a+b x^2\right )^2}-\frac {\sin (c+d x)}{4 b^2 \left (a+b x^2\right )}+\frac {d^2 \cos \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 b^3}+\frac {3 d \sin \left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{16 \sqrt {-a} b^{5/2}}-\frac {d^2 \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 b^3}+\frac {3 d \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{16 \sqrt {-a} b^{5/2}} \]
-1/8*d*x*cos(d*x+c)/b^2/(b*x^2+a)-1/16*d^2*cos(c+d*(-a)^(1/2)/b^(1/2))*Si( d*x-d*(-a)^(1/2)/b^(1/2))/b^3-1/16*d^2*cos(c-d*(-a)^(1/2)/b^(1/2))*Si(d*x+ d*(-a)^(1/2)/b^(1/2))/b^3-1/4*x^2*sin(d*x+c)/b/(b*x^2+a)^2-1/4*sin(d*x+c)/ b^2/(b*x^2+a)-1/16*d^2*Ci(d*x+d*(-a)^(1/2)/b^(1/2))*sin(c-d*(-a)^(1/2)/b^( 1/2))/b^3-1/16*d^2*Ci(-d*x+d*(-a)^(1/2)/b^(1/2))*sin(c+d*(-a)^(1/2)/b^(1/2 ))/b^3-3/16*d*Ci(d*x+d*(-a)^(1/2)/b^(1/2))*cos(c-d*(-a)^(1/2)/b^(1/2))/b^( 5/2)/(-a)^(1/2)+3/16*d*Ci(-d*x+d*(-a)^(1/2)/b^(1/2))*cos(c+d*(-a)^(1/2)/b^ (1/2))/b^(5/2)/(-a)^(1/2)+3/16*d*Si(d*x+d*(-a)^(1/2)/b^(1/2))*sin(c-d*(-a) ^(1/2)/b^(1/2))/b^(5/2)/(-a)^(1/2)-3/16*d*Si(d*x-d*(-a)^(1/2)/b^(1/2))*sin (c+d*(-a)^(1/2)/b^(1/2))/b^(5/2)/(-a)^(1/2)
Result contains complex when optimal does not.
Time = 2.48 (sec) , antiderivative size = 330, normalized size of antiderivative = 0.69 \[ \int \frac {x^3 \sin (c+d x)}{\left (a+b x^2\right )^3} \, dx=\frac {-\frac {i d e^{-i c-\frac {\sqrt {a} d}{\sqrt {b}}} \left (\left (3 \sqrt {b}+\sqrt {a} d\right ) e^{\frac {2 \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )+\left (-3 \sqrt {b}+\sqrt {a} d\right ) \operatorname {ExpIntegralEi}\left (\frac {\sqrt {a} d}{\sqrt {b}}-i d x\right )\right )}{\sqrt {a}}+\frac {i d e^{i c-\frac {\sqrt {a} d}{\sqrt {b}}} \left (\left (3 \sqrt {b}+\sqrt {a} d\right ) e^{\frac {2 \sqrt {a} d}{\sqrt {b}}} \operatorname {ExpIntegralEi}\left (-\frac {\sqrt {a} d}{\sqrt {b}}+i d x\right )+\left (-3 \sqrt {b}+\sqrt {a} d\right ) \operatorname {ExpIntegralEi}\left (\frac {\sqrt {a} d}{\sqrt {b}}+i d x\right )\right )}{\sqrt {a}}-\frac {4 b \cos (d x) \left (d x \left (a+b x^2\right ) \cos (c)+2 \left (a+2 b x^2\right ) \sin (c)\right )}{\left (a+b x^2\right )^2}+\frac {4 b \left (-2 \left (a+2 b x^2\right ) \cos (c)+d x \left (a+b x^2\right ) \sin (c)\right ) \sin (d x)}{\left (a+b x^2\right )^2}}{32 b^3} \]
(((-I)*d*E^((-I)*c - (Sqrt[a]*d)/Sqrt[b])*((3*Sqrt[b] + Sqrt[a]*d)*E^((2*S qrt[a]*d)/Sqrt[b])*ExpIntegralEi[-((Sqrt[a]*d)/Sqrt[b]) - I*d*x] + (-3*Sqr t[b] + Sqrt[a]*d)*ExpIntegralEi[(Sqrt[a]*d)/Sqrt[b] - I*d*x]))/Sqrt[a] + ( I*d*E^(I*c - (Sqrt[a]*d)/Sqrt[b])*((3*Sqrt[b] + Sqrt[a]*d)*E^((2*Sqrt[a]*d )/Sqrt[b])*ExpIntegralEi[-((Sqrt[a]*d)/Sqrt[b]) + I*d*x] + (-3*Sqrt[b] + S qrt[a]*d)*ExpIntegralEi[(Sqrt[a]*d)/Sqrt[b] + I*d*x]))/Sqrt[a] - (4*b*Cos[ d*x]*(d*x*(a + b*x^2)*Cos[c] + 2*(a + 2*b*x^2)*Sin[c]))/(a + b*x^2)^2 + (4 *b*(-2*(a + 2*b*x^2)*Cos[c] + d*x*(a + b*x^2)*Sin[c])*Sin[d*x])/(a + b*x^2 )^2)/(32*b^3)
Time = 1.81 (sec) , antiderivative size = 714, normalized size of antiderivative = 1.50, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {3824, 3822, 3815, 2009, 3825, 3815, 2009, 3826, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {x^3 \sin (c+d x)}{\left (a+b x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 3824 |
| \(\displaystyle \frac {\int \frac {x \sin (c+d x)}{\left (b x^2+a\right )^2}dx}{2 b}+\frac {d \int \frac {x^2 \cos (c+d x)}{\left (b x^2+a\right )^2}dx}{4 b}-\frac {x^2 \sin (c+d x)}{4 b \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 3822 |
| \(\displaystyle \frac {d \int \frac {x^2 \cos (c+d x)}{\left (b x^2+a\right )^2}dx}{4 b}+\frac {\frac {d \int \frac {\cos (c+d x)}{b x^2+a}dx}{2 b}-\frac {\sin (c+d x)}{2 b \left (a+b x^2\right )}}{2 b}-\frac {x^2 \sin (c+d x)}{4 b \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 3815 |
| \(\displaystyle \frac {d \int \frac {x^2 \cos (c+d x)}{\left (b x^2+a\right )^2}dx}{4 b}+\frac {\frac {d \int \left (\frac {\sqrt {-a} \cos (c+d x)}{2 a \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sqrt {-a} \cos (c+d x)}{2 a \left (\sqrt {b} x+\sqrt {-a}\right )}\right )dx}{2 b}-\frac {\sin (c+d x)}{2 b \left (a+b x^2\right )}}{2 b}-\frac {x^2 \sin (c+d x)}{4 b \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d \int \frac {x^2 \cos (c+d x)}{\left (b x^2+a\right )^2}dx}{4 b}+\frac {\frac {d \left (\frac {\cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 \sqrt {-a} \sqrt {b}}-\frac {\cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 \sqrt {-a} \sqrt {b}}+\frac {\sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 \sqrt {-a} \sqrt {b}}+\frac {\sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 \sqrt {-a} \sqrt {b}}\right )}{2 b}-\frac {\sin (c+d x)}{2 b \left (a+b x^2\right )}}{2 b}-\frac {x^2 \sin (c+d x)}{4 b \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 3825 |
| \(\displaystyle \frac {d \left (-\frac {d \int \frac {x \sin (c+d x)}{b x^2+a}dx}{2 b}+\frac {\int \frac {\cos (c+d x)}{b x^2+a}dx}{2 b}-\frac {x \cos (c+d x)}{2 b \left (a+b x^2\right )}\right )}{4 b}+\frac {\frac {d \left (\frac {\cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 \sqrt {-a} \sqrt {b}}-\frac {\cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 \sqrt {-a} \sqrt {b}}+\frac {\sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 \sqrt {-a} \sqrt {b}}+\frac {\sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 \sqrt {-a} \sqrt {b}}\right )}{2 b}-\frac {\sin (c+d x)}{2 b \left (a+b x^2\right )}}{2 b}-\frac {x^2 \sin (c+d x)}{4 b \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 3815 |
| \(\displaystyle \frac {d \left (-\frac {d \int \frac {x \sin (c+d x)}{b x^2+a}dx}{2 b}+\frac {\int \left (\frac {\sqrt {-a} \cos (c+d x)}{2 a \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\sqrt {-a} \cos (c+d x)}{2 a \left (\sqrt {b} x+\sqrt {-a}\right )}\right )dx}{2 b}-\frac {x \cos (c+d x)}{2 b \left (a+b x^2\right )}\right )}{4 b}+\frac {\frac {d \left (\frac {\cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 \sqrt {-a} \sqrt {b}}-\frac {\cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 \sqrt {-a} \sqrt {b}}+\frac {\sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 \sqrt {-a} \sqrt {b}}+\frac {\sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 \sqrt {-a} \sqrt {b}}\right )}{2 b}-\frac {\sin (c+d x)}{2 b \left (a+b x^2\right )}}{2 b}-\frac {x^2 \sin (c+d x)}{4 b \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d \left (-\frac {d \int \frac {x \sin (c+d x)}{b x^2+a}dx}{2 b}+\frac {\frac {\cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 \sqrt {-a} \sqrt {b}}-\frac {\cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 \sqrt {-a} \sqrt {b}}+\frac {\sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 \sqrt {-a} \sqrt {b}}+\frac {\sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 \sqrt {-a} \sqrt {b}}}{2 b}-\frac {x \cos (c+d x)}{2 b \left (a+b x^2\right )}\right )}{4 b}+\frac {\frac {d \left (\frac {\cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 \sqrt {-a} \sqrt {b}}-\frac {\cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 \sqrt {-a} \sqrt {b}}+\frac {\sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 \sqrt {-a} \sqrt {b}}+\frac {\sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 \sqrt {-a} \sqrt {b}}\right )}{2 b}-\frac {\sin (c+d x)}{2 b \left (a+b x^2\right )}}{2 b}-\frac {x^2 \sin (c+d x)}{4 b \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 3826 |
| \(\displaystyle \frac {d \left (-\frac {d \int \left (\frac {\sin (c+d x)}{2 \sqrt {b} \left (\sqrt {b} x+\sqrt {-a}\right )}-\frac {\sin (c+d x)}{2 \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )}\right )dx}{2 b}+\frac {\frac {\cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 \sqrt {-a} \sqrt {b}}-\frac {\cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 \sqrt {-a} \sqrt {b}}+\frac {\sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 \sqrt {-a} \sqrt {b}}+\frac {\sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 \sqrt {-a} \sqrt {b}}}{2 b}-\frac {x \cos (c+d x)}{2 b \left (a+b x^2\right )}\right )}{4 b}+\frac {\frac {d \left (\frac {\cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 \sqrt {-a} \sqrt {b}}-\frac {\cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 \sqrt {-a} \sqrt {b}}+\frac {\sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 \sqrt {-a} \sqrt {b}}+\frac {\sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 \sqrt {-a} \sqrt {b}}\right )}{2 b}-\frac {\sin (c+d x)}{2 b \left (a+b x^2\right )}}{2 b}-\frac {x^2 \sin (c+d x)}{4 b \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d \left (-\frac {d \left (\frac {\sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 b}+\frac {\sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b}-\frac {\cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 b}+\frac {\cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 b}\right )}{2 b}+\frac {\frac {\cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 \sqrt {-a} \sqrt {b}}-\frac {\cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 \sqrt {-a} \sqrt {b}}+\frac {\sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 \sqrt {-a} \sqrt {b}}+\frac {\sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 \sqrt {-a} \sqrt {b}}}{2 b}-\frac {x \cos (c+d x)}{2 b \left (a+b x^2\right )}\right )}{4 b}+\frac {\frac {d \left (\frac {\cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \operatorname {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 \sqrt {-a} \sqrt {b}}-\frac {\cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \operatorname {CosIntegral}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 \sqrt {-a} \sqrt {b}}+\frac {\sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 \sqrt {-a} \sqrt {b}}+\frac {\sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 \sqrt {-a} \sqrt {b}}\right )}{2 b}-\frac {\sin (c+d x)}{2 b \left (a+b x^2\right )}}{2 b}-\frac {x^2 \sin (c+d x)}{4 b \left (a+b x^2\right )^2}\) |
-1/4*(x^2*Sin[c + d*x])/(b*(a + b*x^2)^2) + (d*(-1/2*(x*Cos[c + d*x])/(b*( a + b*x^2)) - (d*((CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sin[c - (Sqrt[- a]*d)/Sqrt[b]])/(2*b) + (CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x]*Sin[c + ( Sqrt[-a]*d)/Sqrt[b]])/(2*b) - (Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[( Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*b) + (Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*SinInte gral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*b)))/(2*b) + ((Cos[c + (Sqrt[-a]*d)/S qrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*Sqrt[-a]*Sqrt[b]) - (C os[c - (Sqrt[-a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*S qrt[-a]*Sqrt[b]) + (Sin[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d) /Sqrt[b] - d*x])/(2*Sqrt[-a]*Sqrt[b]) + (Sin[c - (Sqrt[-a]*d)/Sqrt[b]]*Sin Integral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*Sqrt[-a]*Sqrt[b]))/(2*b)))/(4*b) + (-1/2*Sin[c + d*x]/(b*(a + b*x^2)) + (d*((Cos[c + (Sqrt[-a]*d)/Sqrt[b]]* CosIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*Sqrt[-a]*Sqrt[b]) - (Cos[c - ( Sqrt[-a]*d)/Sqrt[b]]*CosIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*Sqrt[-a]* Sqrt[b]) + (Sin[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(2*Sqrt[-a]*Sqrt[b]) + (Sin[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral [(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*Sqrt[-a]*Sqrt[b])))/(2*b))/(2*b)
3.1.72.3.1 Defintions of rubi rules used
Int[Cos[(c_.) + (d_.)*(x_)]*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int [ExpandIntegrand[Cos[c + d*x], (a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[p, 0] && IGtQ[n, 0] && (EqQ[n, 2] || EqQ[p, -1])
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*Sin[(c_.) + (d_.)*(x_) ], x_Symbol] :> Simp[e^m*(a + b*x^n)^(p + 1)*(Sin[c + d*x]/(b*n*(p + 1))), x] - Simp[d*(e^m/(b*n*(p + 1))) Int[(a + b*x^n)^(p + 1)*Cos[c + d*x], x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && ILtQ[p, -1] && EqQ[m, n - 1] && ( IntegerQ[n] || GtQ[e, 0])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*Sin[(c_.) + (d_.)*(x_)], x_Sym bol] :> Simp[x^(m - n + 1)*(a + b*x^n)^(p + 1)*(Sin[c + d*x]/(b*n*(p + 1))) , x] + (-Simp[(m - n + 1)/(b*n*(p + 1)) Int[x^(m - n)*(a + b*x^n)^(p + 1) *Sin[c + d*x], x], x] - Simp[d/(b*n*(p + 1)) Int[x^(m - n + 1)*(a + b*x^n )^(p + 1)*Cos[c + d*x], x], x]) /; FreeQ[{a, b, c, d, m}, x] && ILtQ[p, -1] && IGtQ[n, 0] && (GtQ[m - n + 1, 0] || GtQ[n, 2]) && RationalQ[m]
Int[Cos[(c_.) + (d_.)*(x_)]*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Sym bol] :> Simp[x^(m - n + 1)*(a + b*x^n)^(p + 1)*(Cos[c + d*x]/(b*n*(p + 1))) , x] + (-Simp[(m - n + 1)/(b*n*(p + 1)) Int[x^(m - n)*(a + b*x^n)^(p + 1) *Cos[c + d*x], x], x] + Simp[d/(b*n*(p + 1)) Int[x^(m - n + 1)*(a + b*x^n )^(p + 1)*Sin[c + d*x], x], x]) /; FreeQ[{a, b, c, d, m}, x] && ILtQ[p, -1] && IGtQ[n, 0] && (GtQ[m - n + 1, 0] || GtQ[n, 2]) && RationalQ[m]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*Sin[(c_.) + (d_.)*(x_)], x_Sym bol] :> Int[ExpandIntegrand[Sin[c + d*x], x^m*(a + b*x^n)^p, x], x] /; Free Q[{a, b, c, d, m}, x] && ILtQ[p, 0] && IGtQ[n, 0] && (EqQ[n, 2] || EqQ[p, - 1]) && IntegerQ[m]
Result contains complex when optimal does not.
Time = 1.10 (sec) , antiderivative size = 640, normalized size of antiderivative = 1.34
| method | result | size |
| risch | \(-\frac {i d^{2} {\mathrm e}^{\frac {i c b +d \sqrt {a b}}{b}} \operatorname {Ei}_{1}\left (\frac {i c b +d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right )}{32 b^{3}}-\frac {i d^{2} {\mathrm e}^{\frac {i c b -d \sqrt {a b}}{b}} \operatorname {Ei}_{1}\left (-\frac {-i c b +d \sqrt {a b}+b \left (i d x +i c \right )}{b}\right )}{32 b^{3}}-\frac {3 i d \sqrt {a b}\, {\mathrm e}^{\frac {i c b +d \sqrt {a b}}{b}} \operatorname {Ei}_{1}\left (\frac {i c b +d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right )}{32 a \,b^{3}}+\frac {3 i d \sqrt {a b}\, {\mathrm e}^{\frac {i c b -d \sqrt {a b}}{b}} \operatorname {Ei}_{1}\left (-\frac {-i c b +d \sqrt {a b}+b \left (i d x +i c \right )}{b}\right )}{32 a \,b^{3}}+\frac {i d^{2} \operatorname {Ei}_{1}\left (-\frac {i c b +d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right ) {\mathrm e}^{-\frac {i c b +d \sqrt {a b}}{b}}}{32 b^{3}}+\frac {i d^{2} {\mathrm e}^{-\frac {i c b -d \sqrt {a b}}{b}} \operatorname {Ei}_{1}\left (\frac {-i c b +d \sqrt {a b}+b \left (i d x +i c \right )}{b}\right )}{32 b^{3}}-\frac {3 i d \,\operatorname {Ei}_{1}\left (-\frac {i c b +d \sqrt {a b}-b \left (i d x +i c \right )}{b}\right ) \sqrt {a b}\, {\mathrm e}^{-\frac {i c b +d \sqrt {a b}}{b}}}{32 a \,b^{3}}+\frac {3 i d \,{\mathrm e}^{-\frac {i c b -d \sqrt {a b}}{b}} \operatorname {Ei}_{1}\left (\frac {-i c b +d \sqrt {a b}+b \left (i d x +i c \right )}{b}\right ) \sqrt {a b}}{32 a \,b^{3}}+\frac {\left (a b \,d^{5} x^{3}+a^{2} d^{5} x \right ) \cos \left (d x +c \right )}{8 a \,b^{2} \left (-b^{2} x^{4} d^{4}-2 a b \,d^{4} x^{2}-a^{2} d^{4}\right )}-\frac {\left (-4 a^{2} b \,d^{6} x^{2}-2 a^{3} d^{6}\right ) \sin \left (d x +c \right )}{8 d^{2} a^{2} b^{2} \left (-b^{2} x^{4} d^{4}-2 a b \,d^{4} x^{2}-a^{2} d^{4}\right )}\) | \(640\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(3353\) |
| default | \(\text {Expression too large to display}\) | \(3353\) |
-1/32*I*d^2/b^3*exp((I*c*b+d*(a*b)^(1/2))/b)*Ei(1,(I*c*b+d*(a*b)^(1/2)-b*( I*d*x+I*c))/b)-1/32*I*d^2/b^3*exp((I*c*b-d*(a*b)^(1/2))/b)*Ei(1,-(-I*c*b+d *(a*b)^(1/2)+b*(I*d*x+I*c))/b)-3/32*I*d/a/b^3*(a*b)^(1/2)*exp((I*c*b+d*(a* b)^(1/2))/b)*Ei(1,(I*c*b+d*(a*b)^(1/2)-b*(I*d*x+I*c))/b)+3/32*I*d/a/b^3*(a *b)^(1/2)*exp((I*c*b-d*(a*b)^(1/2))/b)*Ei(1,-(-I*c*b+d*(a*b)^(1/2)+b*(I*d* x+I*c))/b)+1/32*I*d^2/b^3*Ei(1,-(I*c*b+d*(a*b)^(1/2)-b*(I*d*x+I*c))/b)*exp (-(I*c*b+d*(a*b)^(1/2))/b)+1/32*I*d^2/b^3*exp(-(I*c*b-d*(a*b)^(1/2))/b)*Ei (1,(-I*c*b+d*(a*b)^(1/2)+b*(I*d*x+I*c))/b)-3/32*I*d/a/b^3*Ei(1,-(I*c*b+d*( a*b)^(1/2)-b*(I*d*x+I*c))/b)*(a*b)^(1/2)*exp(-(I*c*b+d*(a*b)^(1/2))/b)+3/3 2*I*d/a/b^3*exp(-(I*c*b-d*(a*b)^(1/2))/b)*Ei(1,(-I*c*b+d*(a*b)^(1/2)+b*(I* d*x+I*c))/b)*(a*b)^(1/2)+1/8/a*(a*b*d^5*x^3+a^2*d^5*x)/b^2/(-b^2*d^4*x^4-2 *a*b*d^4*x^2-a^2*d^4)*cos(d*x+c)-1/8/d^2*(-4*a^2*b*d^6*x^2-2*a^3*d^6)/a^2/ b^2/(-b^2*d^4*x^4-2*a*b*d^4*x^2-a^2*d^4)*sin(d*x+c)
Result contains complex when optimal does not.
Time = 0.30 (sec) , antiderivative size = 492, normalized size of antiderivative = 1.03 \[ \int \frac {x^3 \sin (c+d x)}{\left (a+b x^2\right )^3} \, dx=-\frac {{\left (-i \, a b^{2} d^{2} x^{4} - 2 i \, a^{2} b d^{2} x^{2} - i \, a^{3} d^{2} + 3 \, {\left (-i \, b^{3} x^{4} - 2 i \, a b^{2} x^{2} - i \, a^{2} b\right )} \sqrt {\frac {a d^{2}}{b}}\right )} {\rm Ei}\left (i \, d x - \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (i \, c + \sqrt {\frac {a d^{2}}{b}}\right )} + {\left (-i \, a b^{2} d^{2} x^{4} - 2 i \, a^{2} b d^{2} x^{2} - i \, a^{3} d^{2} + 3 \, {\left (i \, b^{3} x^{4} + 2 i \, a b^{2} x^{2} + i \, a^{2} b\right )} \sqrt {\frac {a d^{2}}{b}}\right )} {\rm Ei}\left (i \, d x + \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (i \, c - \sqrt {\frac {a d^{2}}{b}}\right )} + {\left (i \, a b^{2} d^{2} x^{4} + 2 i \, a^{2} b d^{2} x^{2} + i \, a^{3} d^{2} + 3 \, {\left (i \, b^{3} x^{4} + 2 i \, a b^{2} x^{2} + i \, a^{2} b\right )} \sqrt {\frac {a d^{2}}{b}}\right )} {\rm Ei}\left (-i \, d x - \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (-i \, c + \sqrt {\frac {a d^{2}}{b}}\right )} + {\left (i \, a b^{2} d^{2} x^{4} + 2 i \, a^{2} b d^{2} x^{2} + i \, a^{3} d^{2} + 3 \, {\left (-i \, b^{3} x^{4} - 2 i \, a b^{2} x^{2} - i \, a^{2} b\right )} \sqrt {\frac {a d^{2}}{b}}\right )} {\rm Ei}\left (-i \, d x + \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (-i \, c - \sqrt {\frac {a d^{2}}{b}}\right )} + 4 \, {\left (a b^{2} d x^{3} + a^{2} b d x\right )} \cos \left (d x + c\right ) + 8 \, {\left (2 \, a b^{2} x^{2} + a^{2} b\right )} \sin \left (d x + c\right )}{32 \, {\left (a b^{5} x^{4} + 2 \, a^{2} b^{4} x^{2} + a^{3} b^{3}\right )}} \]
-1/32*((-I*a*b^2*d^2*x^4 - 2*I*a^2*b*d^2*x^2 - I*a^3*d^2 + 3*(-I*b^3*x^4 - 2*I*a*b^2*x^2 - I*a^2*b)*sqrt(a*d^2/b))*Ei(I*d*x - sqrt(a*d^2/b))*e^(I*c + sqrt(a*d^2/b)) + (-I*a*b^2*d^2*x^4 - 2*I*a^2*b*d^2*x^2 - I*a^3*d^2 + 3*( I*b^3*x^4 + 2*I*a*b^2*x^2 + I*a^2*b)*sqrt(a*d^2/b))*Ei(I*d*x + sqrt(a*d^2/ b))*e^(I*c - sqrt(a*d^2/b)) + (I*a*b^2*d^2*x^4 + 2*I*a^2*b*d^2*x^2 + I*a^3 *d^2 + 3*(I*b^3*x^4 + 2*I*a*b^2*x^2 + I*a^2*b)*sqrt(a*d^2/b))*Ei(-I*d*x - sqrt(a*d^2/b))*e^(-I*c + sqrt(a*d^2/b)) + (I*a*b^2*d^2*x^4 + 2*I*a^2*b*d^2 *x^2 + I*a^3*d^2 + 3*(-I*b^3*x^4 - 2*I*a*b^2*x^2 - I*a^2*b)*sqrt(a*d^2/b)) *Ei(-I*d*x + sqrt(a*d^2/b))*e^(-I*c - sqrt(a*d^2/b)) + 4*(a*b^2*d*x^3 + a^ 2*b*d*x)*cos(d*x + c) + 8*(2*a*b^2*x^2 + a^2*b)*sin(d*x + c))/(a*b^5*x^4 + 2*a^2*b^4*x^2 + a^3*b^3)
Timed out. \[ \int \frac {x^3 \sin (c+d x)}{\left (a+b x^2\right )^3} \, dx=\text {Timed out} \]
\[ \int \frac {x^3 \sin (c+d x)}{\left (a+b x^2\right )^3} \, dx=\int { \frac {x^{3} \sin \left (d x + c\right )}{{\left (b x^{2} + a\right )}^{3}} \,d x } \]
-1/2*(3*(cos(c)^2 + sin(c)^2)*d*x^2*sin(d*x + c) + ((d^2*x^3*cos(c) - 3*d* x^2*sin(c) - 12*x*cos(c))*cos(d*x + c)^2 + (d^2*x^3*cos(c) - 3*d*x^2*sin(c ) - 12*x*cos(c))*sin(d*x + c)^2)*cos(d*x + 2*c) + ((cos(c)^2 + sin(c)^2)*d ^2*x^3 - 12*(cos(c)^2 + sin(c)^2)*x)*cos(d*x + c) - 2*(((b^3*cos(c)^2 + b^ 3*sin(c)^2)*d^3*x^6 + 3*(a*b^2*cos(c)^2 + a*b^2*sin(c)^2)*d^3*x^4 + 3*(a^2 *b*cos(c)^2 + a^2*b*sin(c)^2)*d^3*x^2 + (a^3*cos(c)^2 + a^3*sin(c)^2)*d^3) *cos(d*x + c)^2 + ((b^3*cos(c)^2 + b^3*sin(c)^2)*d^3*x^6 + 3*(a*b^2*cos(c) ^2 + a*b^2*sin(c)^2)*d^3*x^4 + 3*(a^2*b*cos(c)^2 + a^2*b*sin(c)^2)*d^3*x^2 + (a^3*cos(c)^2 + a^3*sin(c)^2)*d^3)*sin(d*x + c)^2)*integrate(3*(3*a*d*x *sin(d*x + c) + ((a*d^2 + 10*b)*x^2 - 2*a)*cos(d*x + c))/(b^4*d^3*x^8 + 4* a*b^3*d^3*x^6 + 6*a^2*b^2*d^3*x^4 + 4*a^3*b*d^3*x^2 + a^4*d^3), x) - 2*((( b^3*cos(c)^2 + b^3*sin(c)^2)*d^3*x^6 + 3*(a*b^2*cos(c)^2 + a*b^2*sin(c)^2) *d^3*x^4 + 3*(a^2*b*cos(c)^2 + a^2*b*sin(c)^2)*d^3*x^2 + (a^3*cos(c)^2 + a ^3*sin(c)^2)*d^3)*cos(d*x + c)^2 + ((b^3*cos(c)^2 + b^3*sin(c)^2)*d^3*x^6 + 3*(a*b^2*cos(c)^2 + a*b^2*sin(c)^2)*d^3*x^4 + 3*(a^2*b*cos(c)^2 + a^2*b* sin(c)^2)*d^3*x^2 + (a^3*cos(c)^2 + a^3*sin(c)^2)*d^3)*sin(d*x + c)^2)*int egrate(3*(3*a*d*x*sin(d*x + c) + ((a*d^2 + 10*b)*x^2 - 2*a)*cos(d*x + c))/ ((b^4*d^3*x^8 + 4*a*b^3*d^3*x^6 + 6*a^2*b^2*d^3*x^4 + 4*a^3*b*d^3*x^2 + a^ 4*d^3)*cos(d*x + c)^2 + (b^4*d^3*x^8 + 4*a*b^3*d^3*x^6 + 6*a^2*b^2*d^3*x^4 + 4*a^3*b*d^3*x^2 + a^4*d^3)*sin(d*x + c)^2), x) + ((d^2*x^3*sin(c) + ...
\[ \int \frac {x^3 \sin (c+d x)}{\left (a+b x^2\right )^3} \, dx=\int { \frac {x^{3} \sin \left (d x + c\right )}{{\left (b x^{2} + a\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {x^3 \sin (c+d x)}{\left (a+b x^2\right )^3} \, dx=\int \frac {x^3\,\sin \left (c+d\,x\right )}{{\left (b\,x^2+a\right )}^3} \,d x \]